Thanks Josh
The Concept of Aiki is like a fine "mystery," only in that everything was revealed.
There is no division between the concept of "aiki".in the various examples you cited
a) the Edo period martial arts texts where, "aiki" (相気) is used to refer to the difficult state of engaging in attack and defense when in the same kisetsu* as the opponent.
b) The use of "aiki" (合気) found in many martial arts writings of the 1800s, with the same meaning as in the Touka Mondou. ( you could have included Tesshu's Muto ryu and Yagyu Shingan ryu here)
c) The meaning in the 1892 "Budo Hiketsu Aiki no Jutsu" (The Secret Budo Techniques of Aiki); here "aiki" is an inner teaching of budo, with the meaning of "being one step ahead of the enemy" (敵より一歩先んずる). Here, "techniques of reading the mind of the enemy" (敵人読心の術) and "the aiki of battle cries" (掛け声の合気) are explained with "being ahead" (先んずる) as a presupposition, but specifics are not noted.
d) Sawaga Nenokichi, note to "apply aiki here." often wrote in some 1913 notes, "apply aiki" (合気をかける)

There is also no division when you look at Ueshiba.-although I believe you have the order confused. You stated:However, as Ueshiba-ryu spread, and as Ueshiba's students and successors sought to compensate for this vagueness, interpretations such as this came about: as "aiki" is made up of the characters for "ai" and "ki", it is interpreted as "a way of joining with the energy (ki) of heaven and earth", or alternatively, a "unity of the heavenly and the human" through the matching of rhythm of movement, or natural movement born of an experiential state of understanding.
It wasn't the students to stated it was "the unity of heaven/ earth/ man" That was Ueshiba. And it is perfectly in line with Aiki in yo ho of Daito ryu.
It was the students who misinterpreted that as a rhythm of movement...etc.

The concept of aiki is one of joining the energy in oneself; gravity (heaven) and up energy (earth) and learning to control and manipulate that through in the center through intent (man), then and only then, when one is in full control of himself can one join with and control the energy of another. This creates sometimes really weird and profound controls of another on the physical level as it feels like you can read people and change what they themselves were trying to do to you and they are left feeling you are in control in spite of their best efforts. This makes a case for the many things written from the physical to the mental, and for the confusion of cause and effect from observers and students alike.
I would make a case that the physical effects are concrete and work across the board. The mental aspects and their effect on an opponent are real albeit it is interesting to note that the mental effects are far more pronounced on people used to the physical effects (students). The documented physical effects of real aiki when joined with decades of intense martial practice made it "feel like" the guy was reading your mind...and that when joined with the unusual physical effects made it seem otherworldly.
The mental effects can be likened to reading moves reading intent of others. That skill needs to be seen for what it is, as other great fighters with no aiki development also exhibited it. It is a natural occurence as your mind relaxes when you get used to fighting and stress.
For those with both skills it was an incredible daunting experience...hence the overstating by some writers.

In any event, You suggest that the non-specificity of the word "aiki" ...led to the vagueness of the interpretation of "aiki" used by Ueshiba. I contend his vagueness was intentional. This becomes evident or was revealed upon examining Ueshiba's peers in Daito ryu. It is clear that aiki was indeed capable of being transmitted. It was made equally clear that it was intentionally withheld by decree from all but a few. Sagawa went on record that the "key" was to train with body in tanren and ....that Takeda told him never to reveal it and to never teach Gaijin.
Then again we also now know that Tokimune tried to teach the tanren and he stated to another Shihan who was complaining that his contemporaries in the Takumakai didn't want them, they wanted waza, that "None of my guys wanted to do them either."

Speaking of weird Japanese expressions in Daito ryu....We have some interesting discussions of a weird phrase Oshikiuchi..."indoor"..."inside the threshold." attached to what Takeda said he learned from Tanamo as aiki!

Funny,, here we are in the 21st century and recently a Shihan in a koryu went on record that the inner teaching of his were now going to be "reserved for the Japanese."
Then we have some 20th centruy discussions of the Chinese arts where internal power was openly "held back."
Cheers
Dan